p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.69(C4×D4), C2.9(C4×Q16), C4⋊C4.214D4, Q8⋊C4⋊5C4, (C2×C4).49Q16, C2.3(C4⋊2Q16), (C22×C4).698D4, C23.783(C2×D4), C22.165(C4×D4), C22.35(C2×Q16), C2.5(D4.2D4), C4.27(C4.4D4), C22.68(C4○D8), (C22×C8).51C22, C4.32(C42⋊2C2), C4.39(C42⋊C2), C22.86(C8⋊C22), C22.4Q16.12C2, (C2×C42).299C22, C2.15(SD16⋊C4), (C22×Q8).28C22, C22.127(C4⋊D4), (C22×C4).1382C23, C2.7(C23.20D4), C2.5(C23.48D4), C22.75(C8.C22), C23.67C23.7C2, C22.7C42.22C2, C2.13(C24.C22), C22.95(C22.D4), (C4×C4⋊C4).19C2, (C2×C8).42(C2×C4), (C2×C2.D8).7C2, C4⋊C4.152(C2×C4), (C2×Q8).80(C2×C4), (C2×C4).1012(C2×D4), (C2×Q8⋊C4).5C2, (C2×C4).578(C4○D4), (C2×C4⋊C4).776C22, (C2×C4).400(C22×C4), SmallGroup(128,660)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.(C4×Q16)
G = < a,b,c,d | a2=b4=c8=1, d2=c4, cbc-1=dbd-1=ab=ba, ac=ca, ad=da, dcd-1=ac-1 >
Subgroups: 244 in 127 conjugacy classes, 58 normal (44 characteristic)
C1, C2, C4, C4, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, Q8⋊C4, Q8⋊C4, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C22×C8, C22×Q8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.67C23, C2×Q8⋊C4, C2×C2.D8, C2.(C4×Q16)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, Q16, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×Q16, C4○D8, C8⋊C22, C8.C22, C24.C22, C4×Q16, SD16⋊C4, C4⋊2Q16, D4.2D4, C23.48D4, C23.20D4, C2.(C4×Q16)
►Regular action on 128 points
Generators in S128
(1 128)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 105)(16 106)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)(65 117)(66 118)(67 119)(68 120)(69 113)(70 114)(71 115)(72 116)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 81)(80 82)(89 100)(90 101)(91 102)(92 103)(93 104)(94 97)(95 98)(96 99)
(1 31 113 79)(2 20 114 82)(3 25 115 73)(4 22 116 84)(5 27 117 75)(6 24 118 86)(7 29 119 77)(8 18 120 88)(9 98 51 43)(10 96 52 38)(11 100 53 45)(12 90 54 40)(13 102 55 47)(14 92 56 34)(15 104 49 41)(16 94 50 36)(17 67 87 126)(19 69 81 128)(21 71 83 122)(23 65 85 124)(26 72 74 123)(28 66 76 125)(30 68 78 127)(32 70 80 121)(33 111 91 59)(35 105 93 61)(37 107 95 63)(39 109 89 57)(42 106 97 62)(44 108 99 64)(46 110 101 58)(48 112 103 60)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 62 5 58)(2 49 6 53)(3 60 7 64)(4 55 8 51)(9 116 13 120)(10 71 14 67)(11 114 15 118)(12 69 16 65)(17 44 21 48)(18 37 22 33)(19 42 23 46)(20 35 24 39)(25 34 29 38)(26 47 30 43)(27 40 31 36)(28 45 32 41)(50 124 54 128)(52 122 56 126)(57 121 61 125)(59 127 63 123)(66 109 70 105)(68 107 72 111)(73 92 77 96)(74 102 78 98)(75 90 79 94)(76 100 80 104)(81 97 85 101)(82 93 86 89)(83 103 87 99)(84 91 88 95)(106 117 110 113)(108 115 112 119)
G:=sub<Sym(128)| (1,128)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(65,117)(66,118)(67,119)(68,120)(69,113)(70,114)(71,115)(72,116)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,81)(80,82)(89,100)(90,101)(91,102)(92,103)(93,104)(94,97)(95,98)(96,99), (1,31,113,79)(2,20,114,82)(3,25,115,73)(4,22,116,84)(5,27,117,75)(6,24,118,86)(7,29,119,77)(8,18,120,88)(9,98,51,43)(10,96,52,38)(11,100,53,45)(12,90,54,40)(13,102,55,47)(14,92,56,34)(15,104,49,41)(16,94,50,36)(17,67,87,126)(19,69,81,128)(21,71,83,122)(23,65,85,124)(26,72,74,123)(28,66,76,125)(30,68,78,127)(32,70,80,121)(33,111,91,59)(35,105,93,61)(37,107,95,63)(39,109,89,57)(42,106,97,62)(44,108,99,64)(46,110,101,58)(48,112,103,60), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,62,5,58)(2,49,6,53)(3,60,7,64)(4,55,8,51)(9,116,13,120)(10,71,14,67)(11,114,15,118)(12,69,16,65)(17,44,21,48)(18,37,22,33)(19,42,23,46)(20,35,24,39)(25,34,29,38)(26,47,30,43)(27,40,31,36)(28,45,32,41)(50,124,54,128)(52,122,56,126)(57,121,61,125)(59,127,63,123)(66,109,70,105)(68,107,72,111)(73,92,77,96)(74,102,78,98)(75,90,79,94)(76,100,80,104)(81,97,85,101)(82,93,86,89)(83,103,87,99)(84,91,88,95)(106,117,110,113)(108,115,112,119)>;
G:=Group( (1,128)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(65,117)(66,118)(67,119)(68,120)(69,113)(70,114)(71,115)(72,116)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,81)(80,82)(89,100)(90,101)(91,102)(92,103)(93,104)(94,97)(95,98)(96,99), (1,31,113,79)(2,20,114,82)(3,25,115,73)(4,22,116,84)(5,27,117,75)(6,24,118,86)(7,29,119,77)(8,18,120,88)(9,98,51,43)(10,96,52,38)(11,100,53,45)(12,90,54,40)(13,102,55,47)(14,92,56,34)(15,104,49,41)(16,94,50,36)(17,67,87,126)(19,69,81,128)(21,71,83,122)(23,65,85,124)(26,72,74,123)(28,66,76,125)(30,68,78,127)(32,70,80,121)(33,111,91,59)(35,105,93,61)(37,107,95,63)(39,109,89,57)(42,106,97,62)(44,108,99,64)(46,110,101,58)(48,112,103,60), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,62,5,58)(2,49,6,53)(3,60,7,64)(4,55,8,51)(9,116,13,120)(10,71,14,67)(11,114,15,118)(12,69,16,65)(17,44,21,48)(18,37,22,33)(19,42,23,46)(20,35,24,39)(25,34,29,38)(26,47,30,43)(27,40,31,36)(28,45,32,41)(50,124,54,128)(52,122,56,126)(57,121,61,125)(59,127,63,123)(66,109,70,105)(68,107,72,111)(73,92,77,96)(74,102,78,98)(75,90,79,94)(76,100,80,104)(81,97,85,101)(82,93,86,89)(83,103,87,99)(84,91,88,95)(106,117,110,113)(108,115,112,119) );
G=PermutationGroup([[(1,128),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,105),(16,106),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60),(65,117),(66,118),(67,119),(68,120),(69,113),(70,114),(71,115),(72,116),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,81),(80,82),(89,100),(90,101),(91,102),(92,103),(93,104),(94,97),(95,98),(96,99)], [(1,31,113,79),(2,20,114,82),(3,25,115,73),(4,22,116,84),(5,27,117,75),(6,24,118,86),(7,29,119,77),(8,18,120,88),(9,98,51,43),(10,96,52,38),(11,100,53,45),(12,90,54,40),(13,102,55,47),(14,92,56,34),(15,104,49,41),(16,94,50,36),(17,67,87,126),(19,69,81,128),(21,71,83,122),(23,65,85,124),(26,72,74,123),(28,66,76,125),(30,68,78,127),(32,70,80,121),(33,111,91,59),(35,105,93,61),(37,107,95,63),(39,109,89,57),(42,106,97,62),(44,108,99,64),(46,110,101,58),(48,112,103,60)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,62,5,58),(2,49,6,53),(3,60,7,64),(4,55,8,51),(9,116,13,120),(10,71,14,67),(11,114,15,118),(12,69,16,65),(17,44,21,48),(18,37,22,33),(19,42,23,46),(20,35,24,39),(25,34,29,38),(26,47,30,43),(27,40,31,36),(28,45,32,41),(50,124,54,128),(52,122,56,126),(57,121,61,125),(59,127,63,123),(66,109,70,105),(68,107,72,111),(73,92,77,96),(74,102,78,98),(75,90,79,94),(76,100,80,104),(81,97,85,101),(82,93,86,89),(83,103,87,99),(84,91,88,95),(106,117,110,113),(108,115,112,119)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4R | 4S | 4T | 4U | 4V | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q16 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | C2.(C4×Q16) | C22.7C42 | C22.4Q16 | C4×C4⋊C4 | C23.67C23 | C2×Q8⋊C4 | C2×C2.D8 | Q8⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 4 | 8 | 4 | 1 | 1 |
Matrix representation of C2.(C4×Q16) ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 11 | 7 | 0 | 0 | 0 | 0 |
| 12 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 13 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,13,0,0,0,0,8,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,3,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,6,3,0,0,0,0,11,0],[11,12,0,0,0,0,7,6,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,13] >;
C2.(C4×Q16) in GAP, Magma, Sage, TeX
C_2.(C_4\times Q_{16}) % in TeX
G:=Group("C2.(C4xQ16)"); // GroupNames label
G:=SmallGroup(128,660);
// by ID
G=gap.SmallGroup(128,660);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,394,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,c*b*c^-1=d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=a*c^-1>;
// generators/relations